Best Known (226−40, 226, s)-Nets in Base 2
(226−40, 226, 380)-Net over F2 — Constructive and digital
Digital (186, 226, 380)-net over F2, using
- 21 times duplication [i] based on digital (185, 225, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 45, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 45, 76)-net over F32, using
(226−40, 226, 921)-Net over F2 — Digital
Digital (186, 226, 921)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2226, 921, F2, 2, 40) (dual of [(921, 2), 1616, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 1038, F2, 2, 40) (dual of [(1038, 2), 1850, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2226, 2076, F2, 40) (dual of [2076, 1850, 41]-code), using
- 1 times truncation [i] based on linear OA(2227, 2077, F2, 41) (dual of [2077, 1850, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(2221, 2049, F2, 41) (dual of [2049, 1828, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2199, 2049, F2, 37) (dual of [2049, 1850, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- 1 times truncation [i] based on linear OA(2227, 2077, F2, 41) (dual of [2077, 1850, 42]-code), using
- OOA 2-folding [i] based on linear OA(2226, 2076, F2, 40) (dual of [2076, 1850, 41]-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 1038, F2, 2, 40) (dual of [(1038, 2), 1850, 41]-NRT-code), using
(226−40, 226, 20909)-Net in Base 2 — Upper bound on s
There is no (186, 226, 20910)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 107 942799 519997 351632 739625 706120 594849 770552 131844 686769 687139 431318 > 2226 [i]